Optimal Uncertainty Relations for a Single Observable

Abstract

Uncertainty relations are usually formulated as trade-offs between two or more observables. Yet the uncertainty of a single observable can already contain an intrinsically quantum contribution arising from its noncommutativity with the state. In the sense of Luo, such a contribution can be quantified by a quantum uncertainty $Q_ρ(A)$, with prominent examples including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed. We solve this fixed-state optimization problem by determining the largest coefficient $c_{\rm opt}(ρ)$ for which $V_ρ(A)\ge c_{\rm opt}(ρ)Q_ρ(A)$ holds for every observable $A$, and show that it depends only on the smallest and largest eigenvalues of $ρ$. The same optimization principle extends to a new power-commutator family $\frac{1}{2}|[ρ^s,A]|^2$ $(s\ge 1/2)$, providing a direct measure of state--observable noncommutativity beyond the standard quantum-uncertainty framework. More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.